On Symmetry Through-put linking 3-4-5-6-7 Dimension's ,
Octo-Symmetry for the { 4-Dimension Quasi-Octahedron } ,
and 2 Identical { 8-D Structures } oriented Symmetrically

by Frank C. Fung ( 1st published in July, 2017. )

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Section XIX - Setting up the 4 Sets of { 8-D Orthogonal Unit-Vector Basis } for our Analysis

Summary for the Section :

• In this Section , we shall set up :
  • 4 sets of { 8-D orthogonal unit-vector basis } ,

    for use in our analysis here in Part IV of the paper .

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Defining the { 8-Dimension Vector Space } :

• For our analysis purposes here in Part IV ,
  • it is necessary for us to define , at this first-instance ,
    • the { 8-Dimension Vector Space } via 8 [ Physical Unit-Vectors ] .

  We shall now identify a set of eight (8) unit-vectors in the { 8-Dimension Vector Space } , so that :

  • the 8 unit-vectors are orthogonal to one-another .

  For the remainder of this Part IV here ,

  • we shall treat these { 8 unit-vectors } as fixed-and-invariant in the { 8-Dimension Vector Space } ,

    and

  • we shall name these 8 [ physical unit-vectors ] :
    • { Vector PUV-1 } , { Vector PUV-2 } , { Vector PUV-3 } , { Vector PUV-4 } ,
      { Vector PUV-5 } , { Vector PUV-6 } , { Vector PUV-7 } and { Vector PUV-8 } ;

      respectively .

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Setting up { Orthogonal Unit-Vector Basis SET ZERO } :

• Let us now set up the first of 4 sets of { orthogonal unit-vector basis } , namely :
  • { Orthogonal Unit-Vector Basis SET ZERO } .

    Details for the Orthogonal Unit-Vector Basis { SET ZERO }
    Vector I.D.	* *	Co-ordinates based on the Physical Unit Vector
    		PUV-1	PUV-2	PUV-3	PUV-4	PUV-5	PUV-6	PUV-7	PUV-8
    *************	*	******	******	******	******	******	******	******	******
    Vector UVZ-1	*	+1	0	0	0	0	0	0	0
    Vector UVZ-2	*	0	+1	0	0	0	0	0	0
    Vector UVZ-3	*	0	0	+1	0	0	0	0	0
    Vector UVZ-4	*	0	0	0	+1	0	0	0	0
    Vector UVZ-5	*	0	0	0	0	+1	0	0	0
    Vector UVZ-6	*	0	0	0	0	0	+1	0	0
    Vector UVZ-7	*	0	0	0	0	0	0	+1	0
    Vector UVZ-8	*	0	0	0	0	0	0	0	+1

  And we shall name the 8 unit-vectors in { SET ZERO } :

  • { Vector UVZ-1 } , { Uector UVZ-2 } , { Vector UVZ-3 } , { Vector UVZ-4 } ,
    { Vector UVZ-5 } , { Uector UVZ-6 } , { Vector UVZ-7 } , { Vector UVZ-8 } ;

    respectively .

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Setting up { Orthogonal Unit-Vector Basis SET ALPHA } :

• Let us now set up the second of 4 sets of { orthogonal unit-vector basis } , namely :
  • { Orthogonal Unit-Vector Basis SET ALPHA } .

    Details for the Orthogonal Unit-Vector Basis { SET ALPHA }
    Vector I.D.	* *	Co-ordinates based on the Physical Unit Vector
    		PUV-1	PUV-2	PUV-3	PUV-4	PUV-5	PUV-6	PUV-7	PUV-8
    *************	*	******	******	******	******	******	******	******	******
    Vector UVA-1	*	+1	0	0	0	0	0	0	0
    Vector UVA-2	*	0	+1	0	0	0	0	0	0
    Vector UVA-3	*	0	0	-1	0	0	0	0	0
    Vector UVA-4	*	0	0	0	+1	0	0	0	0
    Vector UVA-5	*	0	0	0	0	+1	0	0	0
    Vector UVA-6	*	0	0	0	0	0	+1	0	0
    Vector UVA-7	*	0	0	0	0	0	0	+1	0
    Vector UVA-8	*	0	0	0	0	0	0	0	+1

  And we shall name the 8 unit-vectors in { SET ALPHA } :

  • { Vector UVA-1 } , { Uector UVA-2 } , { Vector UVA-3 } , { Vector UVA-4 } ,
    { Vector UVA-5 } , { Uector UVA-6 } , { Vector UVA-7 } , { Vector UVA-8 } ;

    respectively .

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Setting up { Orthogonal Unit-Vector Basis SET OMEGA } :

• Let us now set up the third of 4 sets of { orthogonal unit-vector basis } , namely :
  • { Orthogonal Unit-Vector Basis SET OMEGA } .

    Details for the Orthogonal Unit-Vector Basis { SET OMEGA }
    Vector I.D.	* *	Co-ordinates based on the Physical Unit Vector
    		PUV-1	PUV-2	PUV-3	PUV-4	PUV-5	PUV-6	PUV-7	PUV-8
    *************	*	******	******	******	******	******	******	******	******
    Vector UVO-1	*	-1	0	0	0	0	0	0	0
    Vector UVO-2	*	0	-1	0	0	0	0	0	0
    Vector UVO-3	*	0	0	+1	0	0	0	0	0
    Vector UVA-4	*	0	0	0	-1	0	0	0	0
    Vector UVO-5	*	0	0	0	0	-1	0	0	0
    Vector UVO-6	*	0	0	0	0	0	-1	0	0
    Vector UVO-7	*	0	0	0	0	0	0	-1	0
    Vector UVO-8	*	0	0	0	0	0	0	0	-1

  And we shall name the 8 unit-vectors in { SET OMEGA } :

  • { Vector UVO-1 } , { Uector UVO-2 } , { Vector UVO-3 } , { Vector UVO-4 } ,
    { Vector UVO-5 } , { Uector UVO-6 } , { Vector UVO-7 } , { Vector UVO-8 } ;

    respectively .

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Setting up { Orthogonal Unit-Vector Basis SET RICO } :

• Let us now set up the fourth of 4 sets of { orthogonal unit-vector basis } , namely :
  • { Orthogonal Unit-Vector Basis SET RICO } .

    Details for the Orthogonal Unit-Vector Basis { SET RICO }
    Vector I.D.	* *	Co-ordinates based on the Physical Unit Vector
    		PUV-1	PUV-2	PUV-3	PUV-4	PUV-5	PUV-6	PUV-7	PUV-8
    *************	*	******	******	******	******	******	******	******	******
    Vector UVR-1	*	-1	0	0	0	0	0	0	0
    Vector UVR-2	*	0	-1	0	0	0	0	0	0
    Vector UVR-3	*	0	0	-1	0	0	0	0	0
    Vector UVR-4	*	0	0	0	-1	0	0	0	0
    Vector UVR-5	*	0	0	0	0	-1	0	0	0
    Vector UVR-6	*	0	0	0	0	0	-1	0	0
    Vector UVR-7	*	0	0	0	0	0	0	-1	0
    Vector UVR-8	*	0	0	0	0	0	0	0	-1

  And we shall name the 8 unit-vectors in { SET RICO } :

  • { Vector UVR-1 } , { Uector UVR-2 } , { Vector UVR-3 } , { Vector UVR-4 } ,
    { Vector UVR-5 } , { Uector UVR-6 } , { Vector UVR-7 } , { Vector UVR-8 } ;

    respectively .

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What's Next :

• We shall now proceed to demonstrate :
  • why { Structure A } and { Structure B } can be identical in shape , next .

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go to the next Section : Section XX - Why { Structure A } and { Structure B } are Identical In Shape

go to the last Section : Section XVIII - 2 Identical Structures arising from the { 8-D Quasi-Cube }

return to the Symmetry Through-put HomePage

Original dated 2017-7-23